\newproblem{lay:5_4_3}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.4.3}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $\mathcal{E}=\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\}$ be the standard basis for $\mathbb{R}^3$, let $\mathcal{B}=\{\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\}$ 
	be a basis for a vector space $V$, and let $T:\mathbb{R}^3 \rightarrow V$ be a linear transformation with the property that
	\begin{center}
		$T(x_1,x_2,x_3)=(2x_3-x_2)\mathbf{b}_1-(2x_2)\mathbf{b}_2+(x_1+3x_3)\mathbf{b}_3$
	\end{center}
	\begin{enumerate}[a.]
		\item Compute $T(\mathbf{e}_1)$, $T(\mathbf{e}_2)$ and $T(\mathbf{e}_3)$.
		\item Compute $[T(\mathbf{e}_1)]_{\mathcal{B}}$, $[T(\mathbf{e}_2)]_{\mathcal{B}}$ and $[T(\mathbf{e}_3)]_{\mathcal{B}}$.
		\item Find the matrix for $T$ relative to $\mathcal{E}$ and $\mathcal{B}$
	\end{enumerate}
}{
  % Solution
	\begin{enumerate}[a.]
		\item Applying the transformation $T$ to the three standard vectors we get \\
			    \begin{center}
						$\begin{array}{l}
						   T(\mathbf{e}_1)=T(1,0,0)=\mathbf{b}_3\\
							 T(\mathbf{e}_2)=T(0,1,0)=-\mathbf{b}_1-2\mathbf{b}_2\\
							 T(\mathbf{e}_3)=T(0,0,1)=2\mathbf{b}_1+3\mathbf{b}_3\\
						\end{array}$
					\end{center}
		\item Let's calculate now the coordinates of the different transformed vectors in $\mathcal{B}$
			    \begin{center}
						\begin{tabular}{l}
						   $[T(\mathbf{e}_1)]_{\mathcal{B}}=(0,0,1)$\\
							 $[T(\mathbf{e}_2)]_{\mathcal{B}}=(-1,-2,0)$\\
							 $[T(\mathbf{e}_3)]_{\mathcal{B}}=(2,0,3)$\\
						\end{tabular}
					\end{center}
		\item The matrix sought is the one whose columns are the vectors in part b.
			    \begin{center}
						$M=\begin{pmatrix}0 & -1 & 2 \\ 0 & -2 & 0 \\ 1 & 0 & 3\end{pmatrix}$
					\end{center}
	\end{enumerate}
}
\useproblem{lay:5_4_3}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
